Compute the adjoint of the matrix and verify that A (Adj A) = | A | 1. - Wired Faculty

Question

Compute the adjoint of the matrix $straight A space equals space open square brackets table row 1 cell space space space space space space 2 end cell row 3 cell space space minus 5 end cell end table close square brackets$
and verify that A (Adj A) = | A | 1.

Solution

straight A space equals space open square brackets table row 1 cell space space space space space space space space 2 end cell row 3 cell space space space space minus 5 end cell end table close square brackets

therefore space space space space space open vertical bar straight A close vertical bar space equals space open vertical bar table row 1 cell space space space space space space space 2 end cell row 3 cell space space minus 5 end cell end table close vertical bar space equals space minus 5 minus 6 space equals space minus 11

Co-factor of first element of first row = (– 1)^{1 + 1} (– 5) = – 5
Co-factor of second element of first row = (– 1)^{1 + 2} (3) = – 3
Co-factor of first element of second row = (– 1)^{2 + 1} (2) = –2
Co-factor of second element of second row = (– 1)^{2 + 2} (1) = 1

therefore space space space space space adj. space straight A space equals space open square brackets table row cell negative 5 end cell cell space space space space minus 3 end cell row cell negative 2 end cell cell space space space space space space 1 end cell end table close square brackets space space space open square brackets table row cell negative 5 end cell cell space space space space space minus 2 end cell row cell negative 3 end cell cell space space space space space space 1 end cell end table close square brackets space space space space space space straight A left parenthesis adj. space straight A right parenthesis space space equals space open square brackets table row 1 cell space space space space space space 2 end cell row 3 cell space space minus 5 end cell end table close square brackets space open square brackets table row cell negative 5 end cell cell space space space space minus 2 end cell row cell negative 3 end cell cell space space space space space space space 1 end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space open square brackets table row cell negative 5 minus 6 end cell cell space space space space space space space minus 2 plus 2 end cell row cell negative 15 plus 15 end cell cell space space space space space space space space minus 6 minus 5 end cell end table close square brackets space equals space open square brackets table row cell negative 11 end cell cell space space space 0 end cell row 0 cell negative 11 end cell end table close square brackets space equals space minus 11 open square brackets table row 1 0 row 0 1 end table close square brackets space space space space space space space space space space space space space space space space space space space space space space space equals space open vertical bar straight A close vertical bar space straight I therefore space space space space straight A left parenthesis adj. space straight A right parenthesis space equals space open vertical bar straight A close vertical bar space straight I

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Determinants

Compute the adjoint of the matrix $straight A space equals space open square brackets table row 1 cell space space space space space space 2 end cell row 3 cell space space minus 5 end cell end table close square brackets$
and verify that A (Adj A) = | A | 1.

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